DESIGN AND ANALYSIS OF CAM MECHANISM WITH TRANSLATING FOLLOWER HAVING DOUBLE ROLLERS. Jung-Fa Hsieh

DESIGN AND ANALYSIS OF CAM MECHANISM WITH TRANSLATING FOLLOWER HAVING DOUBLE ROLLERS Jung-Fa Hsieh Far East University Department of Mechanical Engineering Tainan, 74448, Taiwan E-mail: seznof@cc.feu.edu.tw ICETI-2014 J1033_SCI No. 15-CSME-19, E.I.C. Accession 3794 ABSTRACT The pressure angle is one of the primary considerations in designing a cam mechanism since an inappropriate angle may cause excessive sliding loads on the follower. This paper presents a simple yet straightforward method for the design and analysis of a cam mechanism with a translating follower having double rollers. In the proposed approach, conjugate surface theory is employed to derive a kinematic model of the cam mechanism. Analytical expressions for the pressure angle and principal curvatures of the cam profile are then derived. The validity of the analytical expressions is confirmed by machining a designed cam using a 3-axis CNC machine tool. Keywords: homogenous coordinate transformation; double roller; pressure angle. CONCEPTION ET ANALYSE D UN MÉCANISME À CAME AVEC SUIVEURS TRANSLATIONELS À DOULE ROULEAUX RÉSUMÉ L angle de pression est une des premières considérations dans la conception d un mécanisme à came, étant donné qu un angle inapproprié peut causer une charge glissante excessive sur le suiveur. Cet article présente une méthode simple et directe pour la conception et l analyse d un mécanisme à came avec suiveur translationel à double rouleaux. Dans l approche proposée, la théorie de la surface conjuguée est employée pour donner un modèle cinématique du mécanisme à came. Les expressions analytiques pour l angle de pression et les courbes principales du profil de la came sont dérivées. La validité des expressions analytiques est confirmée par la fabrication d une came à l aide d une machine-outil CNC. Mots-clés : transformation homogène coordonnée; double rouleaux; angle de pression. Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015 397

NOMENCLATURE (xyz) 0 Configuration frame (xyz) 0 built in cam 0 S Cam surface r S Roller surface 0 n Unit outward normal of cam surface ρ Roller radius R 0 Pitch circle radius of cam h Life height of follower e Offset amount of roller ψ Pressure angle θ 1 Rotation angle of cam Displacement of follower s 2 1. INTRODUCTION Cam elements have many advantageous features for mechanical transmission systems, including simplicity, reliability, repeatability and a low running noise. However, the pressure angle must be carefully designed since an overly-large angle may result in an excessive sliding load on the follower [1]. In practice, the pressure angle can be reduced by increasing the offset amount of the follower. However, if the offset is specified in such a way as to reduce the pressure angle during the rise stage of the follower, the pressure angle during the fall stage inevitably increases, and vice versa [2]. Hence, a careful design of the offset amount is essential in achieving the optimal compromise. In most practical cam systems, the follower is fitted with a spring which presses it firmly against the cam surface as it rotates. However, Wunderlich [3] proposed a single-disk cam mechanism in which vibration under high-speed conditions was minimized by replacing the spring with an oscillating double roller follower. Duca et al. [4] performed an analytical investigation to establish the design parameters which minimized the pressure angle for the cam mechanism proposed in [3]. Wu et al. [5] presented a disc cam mechanism with a novel translating follower incorporating two double rollers mounted symmetrically on opposite sides of the cam. In developing the proposed mechanism, the instantaneous center of velocity principle was applied to design the cam profile in such a way as to minimize the pressure angle during both the rise stage and the fall stage of the follower motion. Hsieh [6] used a homogeneous coordinate transformation method to derive an analytical expression for the curved slots of a Geneva wheel cam mechanism. A method was then proposed for deriving the corresponding NC machining data. However, the published literature lacks a systematic investigation into the design and machining of a disc cam mechanism with a translating double roller follower. Accordingly, this study presents a simple yet robust methodology for the design, analysis and machining of such mechanisms. The proposed methodology comprises four steps, namely (1) establishing a kinematic model of the cam mechanism, (2) designing the cam profile, (3) analyzing the pressure angle and principal curvatures of the cam, and (4) generating the NC data required to machine the designed cam. The validity of the proposed methodology is demonstrated by machining a designed cam on a 3-axis CNC machine. 2. SURFACE GEOMETRY Figure 1 illustrates the cam mechanism considered in the present study. As shown, the translating follower incorporates two identical rollers, denoted by the suffixes r1 and r2, respectively. The rollers are mounted symmetrically on opposite sides of the follower centerline and have the same offset. As shown in the figure, the cam rotates in the clockwise direction about its center point, 0 0. The profile of the cam comprises four segments, each designed to produce a specific follower motion, namely (1) follower rise (segment p 1 p 2 ), 398 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015

Fig. 1. Cam mechanism with translating follower having double rollers. Table 1. Kinematic parameters of cam mechanism with translating follower having double rollers. b i θ i a i α i Link 1 0 θ 1 0 90 Link 2 b 2 0 0 0 (2) follower dwell (segment p 2 p 3 ), (3) follower fall (segment p 3 p 4 ), and (4) follower dwell (segment p 4 p 0 ). Figure 2 shows a typical double-dwell cam profile. As the cam rotates, the follower rises to a total lift height h over an angle θ r, remains in a high dwell position for the next θ H, falls through a distance h over θ f, and remains in a low dwell position over θ L. As shown, at least one of the rollers remains in contact with the cam surface at all times as the cam rotates. To determine the cam profile using conjugate theory, it is first necessary to label the coordinate frames, starting with the cam (marked as 0 in Fig. 1) and ending with the translating follower (marked as 2 in Fig. 1). Once a frame (xyz) i (i = 0,1,2) has been assigned to every link according to the Denavit Hartenberg (D H) notation [7], the kinematic parameters can be tabulated, as shown in Table 1. Note that parameter b 2 is defined as b 2 = b 20 + s 2 (θ 1 ), where b 20 is the shortest distance between the cam center and the center of the follower rollers as the cam rotates and s 2 is the translating follower displacement. The configuration of frame (xyz) 2 with respect to frame (xyz) 0 is given by 0 A 2 = 2 i=1 i 1 A i = Cθ 1 0 Sθ 1 b 2 Sθ 1 Sθ 1 0 Cθ 1 b 2 Cθ 1 0 1 0 0 0 0 0 1, (1) where C and S denote COSINE and SINE, respectively. To analyze the roller contact with the cam during cam rotation, it is first necessary to establish the frames of the two rollers, namely (xyz) r1 and (xyz) r2, respectively. The configuration of frame (xyz) r1 with respect Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015 399

to frame (xyz) 2 is given by the matrix 2 A r1 = Trans( e,0,0)rot(x 2,90) = 1 0 0 e 0 0 1 0 0 1 0 0 0 0 0 1 where e is the offset amount of two rollers. Similarly, the configuration of frame (xyz) r2 with respect to frame (xyz) 2 is given by 2 A r2 = Trans(e,0,0)Rot(x 2,90) = 1 0 0 e 0 0 1 0 0 1 0 0 0 0 0 1, (2). (3) The two rollers are identical. Thus, the surface equation, r S, and unit outward normal, r n, are also the same for both rollers, and can be expressed with respect to frame (xyz) r as follows: r S = [ρcθ ρsθ u 1] T ( λ/2 u λ/2,0 θ 2π), (4) r n = r S u r S / r S θ u r S θ = [Cθ Sθ 0 0]T, (5) where θ is the polar angle, u is the height parameter of the cylindrical roller, and ρ is the radius of the roller. In accordance with the principles of homogeneous coordinate transformation, the cam surface equation, 0 S, is related to rollers r1 and r2 as follows: 0 S = 0 A 2 2 A r1 r1 S = 0 S = 0 A 2 2 A r1 r1 S = ρc(θ + θ 1 ) ecθ 1 b 2 Sθ 1 ρs(θ + θ 1 ) esθ 1 + b 2 Cθ 1 u 1 ρc(θ + θ 1 ) + ecθ 1 b 2 Sθ 1 ρs(θ + θ 1 ) + esθ 1 + b 2 Cθ 1 u 1 In addition, the normal vector, 0 n, of the cam can be expressed as, (6). (7) o n = 0 A r r n = [C(θ + θ 1 ) S(θ + θ 1 ) 0 0] T. (8) Once the input-output relation of the cam mechanism has been defined, the conjugate points and cam profiles can be determined via the formula 0 n T d0 S dt = 0, (9) where 0 n and 0 S are the unit outward normal and surface equation of the cam profile with respect to frame (xyz) 0, respectively. The conjugate point (denoted as θ) is expressed as ( ) F θ = tan 1, (10) Ē 400 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015

Fig. 2. Cam profile corresponding to roller contact position. where Ē is defined as Ē = b 2, and F is defined as F = e + db 2 /dθ 1 for roller r1 and F = e + db 2 /dθ 1 for roller r2. Consequently, the profiles of the cam corresponding to rollers r1 and r2 can be expressed respectively as 0 S = 0 A r1 r1 S = 0 S = 0 A r1 r1 S = ρc( θ + θ 1 ) ecθ 1 b 2 Sθ 1 ρs( θ + θ 1 ) esθ 1 + b 2 Cθ 1 u 1 ρc( θ + θ 1 ) +Cθ 1 b 2 Sθ 1 ρs( θ + θ 1 ) + esθ 1 + b 2 Cθ 1 u 1, (11). (12) As shown in Fig. 2, the cam profile is determined by roller r1 (i.e., Eq. (11)) over the interval 0 θ 1 (θ r + θ H 2 θ H ) and by roller r2 (i.e., Eq. (12)) over the interval (θ r + θ H ) θ 1 360. As discussed above, the low and high dwell motions of the follower correspond to two particular segments (circular arcs) of the cam profile. For the low dwell interval, the circular cam segment has a radius of R 0, i.e., the base circle radius of the cam. By contrast, for the high dwell interval, the radius R of the circular arc is given by R = (b 20 + h) 2 + e 2 ρ. (13) In the high dwell interval, both of the rollers contact the cam profile. Thus, half the subtending angle of the roller span, θ H can be expressed as ( ) e θ H = sin 1. (14) R + ρ The same condition also applies in the low dwell interval. As a result, half the subtending angle of the roller span, θ L, can be expressed as ( ) e θ L = sin 1. (15) R 0 + ρ Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015 401

Fig. 3. Pressure angle of cam mechanism during: (a) rise of follower and (b) fall of follower. It is noted that to satisfy the conditions given in Eqs. (14) and (15), the two circular arcs corresponding to the high and low dwell intervals, respectively, must be sufficiently large to accommodate the roller span. 3. PRESSURE ANGLE The pressure angle, ψ, provides an important measure of the effectiveness with which a force is transmitted between a cam and a follower. Specifically, a pressure angle of zero degrees maximizes the transmitted force, while a pressure angle of 90 degrees results in a jamming of the follower in its guides. As shown in Fig. 3, the pressure angle is defined as the angle between the direction of motion (i.e., velocity) of the follower and the direction of the force transmission axis [8]. The pressure angle at the contact point p between the cam and the roller is given as tanψ = 1 n p 1 v p 1 n p 1v, (16) p where the unit normal vector of the contact point, 1 n p, and the direction of velocity of the contact point, 1 v p, are both defined with respect to the fixed frame (xyz) 1. The unit normal vector of the contact point on the roller-follower surface with respect to frame (xyz) 1 is given by 1 n p = 1 A 2 2 A r r n p = [C θ 0 S θ 0] T. (17) Meanwhile, the location of the contact point, p, on the roller-follower surface with respect to frame (xyz) 1 is given by 1 S p = 1 A 2 2 A r r S p = [ρc θ e u ρs θ + b 2 1] T. (18) Since the roller-follower translates along the z 1 axis, the direction of velocity of the contact point on the roller-follower surface with respect to frame (xyz) 1 can be expressed as 1 v p = d1 S dt [ = 0 0 db 2 dt Using the inner product and cross product, the pressure angle, ψ, can be derived as 0] T. (19) tanψ = cot θ. (20) 402 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015

In determining the pressure angle, ψ, as the cam rotates, it is necessary to consider the contact status between the two rollers and the cam profile. More specifically: 1. For 0 θ 1 θ r +2 θ H, only roller r1 contacts the cam profile, and the pressure angle can be derived from the conjugate angle θ using Eq. (20). 2. For θ r + 2 θ H θ 1 θ r + θ H 2 θ H and θ r + θ H + θ f θ 1 360, roller r1 and roller r2 both contact the cam profile, and hence the pressure angle can be regarded as zero due to the symmetrical arrangement of the rollers and cam. 3. For θ r + θ H 2 θ H θ 1 θ r + θ H + θ f, only roller r2 contacts the cam profile, and the pressure angle can be derived using Eq. (20). 4. ANALYSIS OF CURVATURE The principal curvatures of the cam surface must be analyzed in order to prevent the occurrence of singular points and to determine the size of the cam surface. In general, the absolute value of the negative radius of curvature of the cam profile should be greater than the radius of the roller such that the roller can fit into the concave part of the cam profile. As a result, the principal curvatures of the cam profile must be carefully designed. According to basic differential geometry principles, the principal curvatures of the cam profile can be evaluated as follows: K 1,K 2 = H ± H 2 K, (21) where K 1 and K 2 are the principal curvatures, and K and H are defined respectively as where K = LN M2 EG F 2 and H = 2FM EN GL 2(EG F 2, (22) ) L = 0 n 2 0 S 2 u, M =0 n 2 0 S u θ 1, N = 0 n 2 0 S 2 θ 1, E = 0 S u 0 S u, F = 0 S u 0 S θ 1, and G = 0 S θ 1 0 S θ 1 (see Eqs. (A1 A6) in Appendix A). Note that the formulas in this appendix correspond to roller r1. For the case of roller r2, the term e should be replaced with e and the term e becomes e. As shown in Eqs. (21) and (22), the principal curvatures of the cam profile are related to both the follower motion program and the dimensions of the kinematic model for the cam mechanism. 5. IMPLEMENTATION To validate the design methodology presented in Sections 2 4, a cam mechanism with a translating follower having double rollers was constructed with design parameters of θ r = 100, θ H = 70, θ f = 130, θ L = 60, e = 10 mm, R 0 = 40 mm and ρ = 8 mm. The follower displacement was specified by a modified sine motion curve, in which θ 1 is the rotational position of the cam, s 2 is the translation displacement of the follower, and θ d is the dwell period. In general, a modified sine motion curve starts from a zero position and rises to a total height h over the period of the cam rotation, τ. In the present study, h was assigned a value of 24 mm. Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015 403

Fig. 4. Simulated profiles of cam given use of symmetric double rollers (e = 10) and conventional single roller (e = 0). Fig. 5. Variation of pressure angle for two different cam profile designs. Moreover, the follower displacement, s2, associated with the modified sine curve was given as follows: ( )] h[ π θ 1 θ d 4+π τ 1 4(4+π) S 4π θ 1 θ d τ,0 θ 1 θ d π 8 ( )] s 2 (θ 1 ) = h[ 2 4+π + π θ 1 θ d 4+π τ 9 4(4+π) S 4π θ 1 θ d 3 τ + π 3, π 8 θ 1 θ d 7τ 8. (23) ( )] h[ 4 4+π + π θ 1 θ d 4+π τ 1 4(4+π) S 4π θ 1 θ d τ, 7τ 8 θ 1 θ d τ Figure 4 shows the simulation results obtained for the cam profile given the use of a symmetric double roller (e = 10) and common single roller (e = 0), respectively. Figure 5 presents the corresponding results for the variation of the pressure angle at the contact point as the cam rotates. It is seen that the symmetric double roller results in a significant reduction in the peak pressure angle. Figure 6 shows the principal 404 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015

Fig. 6. Variation of principal curvatures of designed cam profile with symmetric double rollers. Fig. 7. Photograph of machined cam. curvatures of the cam profile designed for the double-roller follower. It is noted that one of the principal curvatures (i.e., K 1 ) is equal to zero since the cam has a 2D profile. The results indicate that the designed cam has a minimum radius of curvature of 40.0 mm. Thus, the roller maintains continuous contact with the cam. The NC equations required to produce the corresponding cam using a 3-axis machine tool were derived using the methodology described by the present group in [6]. Figure 7 presents a photograph of the machined cam. Note that the three holes lying along the x 0 axis of the cam are the workpiece origin (center hole) and the clamping holes, respectively. It is seen that a good qualitative agreement exists between the machined profile and the designed profile (see Fig. 4). Thus, the feasibility of the proposed methodology is confirmed. 6. CONCLUSIONS This paper has presented an integrated methodology for the design, analysis and machining of a cam mechanism incorporating a translating follower with a double roller. A kinematic model of the cam mechanism has been derived utilizing the homogeneous coordinate transformation method and conjugate surface theory. In addition, analytical expressions have been derived for the pressure angle and principal curvatures of the cam profile. The NC data required to machine the designed cam have been obtained using the method proposed Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015 405

by the present group in a previous study [6]. It has been shown that the profile of the machined cam is in good agreement with the analytically-derived profile. ACKNOWLEDGMENT The author gratefully acknowledges the financial support provided to this study by the National Science Council of Taiwan under Grant No. NSC100-2221-E-269-009. REFERENCES 1. Chen, F.Y., Mechanics and Design of Cam Mechanisms, Pergamon Press, New York, 1982. 2. Norton, R.L., Design of Machinery, McGraw-Hill, New York, 2004. 3. Wunderlich, W., ingle-disk mechanism with oscillating double roller, Mechanism and Machine Theory, Vol. 19, pp. 409 415, 1984 4. Wu, L.I., Liu, C.-H. and Shu, K.-L., Disk cam mechanisms with a translating follower having symmetrical double rollers, Mechanism and Machine Theory, Vol. 44, pp. 2085 2097, 2009. 5. Hsieh, J.-F., Application of homogenous transformation matrix to the design and machining of a Geneva mechanism with curved slots, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 221, pp. 1435 1443, 2007. 6. Hsieh, J.-F., Design and analysis of offset slider-crank with translating roller -follower, Transactions of the Canadian Society for Mechanical Engineering, Vol. 35, pp. 419 436, 2011. 7. Paul, R.P., Robot Manipulators-Mathematics, Programming and Control, MIT Press, Cambridge, MA,1982. 8. Gonzalez-Palacios, M.A. and Angles, J., Cam Synthesis, Springer, 1993. APPENDIX A In deriving the principal curvatures of the cam profile using Eqs. (21) and (22), the related parameter definitions are as follows: L = 0, (A.1) where M = 0, (A.2) ( ) 2 d θ N = r + 1 + ec θ b 2 S θ 2C θ db 2 + S θ d2 b 2 dθ 1 dθ 1 dθ1 2, (A.3) E = 1, (A.4) F = 0, ( ) 2 ( ) ( ) d θ d θ 2 G = r 2 + 1 + (e) 2 + (b 2 ) 2 db2 + 2rC( θ) + 1 + 2rb 2 S( θ) + 2e db 2, dθ 1 dθ 1 dθ 1 dθ 1 d θ dθ 1 = ( ) ( ) e + db 2 db2 dθ 1 dθ 1 b d 2 b 2 2 dθ1 2 b 2 2 + ( e + db)2 dθ 1 ) 2. (A.5) (A.6) 406 Transactions of the Canadian Society for Mechanical Engineering, Vol. 39, No. 3, 2015